06-FMCS-Kananaskis

Computable Real Analysiswithout Set Theory or Turing Machines

Paul Taylor

Department of Computer ScienceUniversity of Manchester

UK EPSRC GR/S58522

Foundational Methods in Computer ScienceKananaskis, Thursday 8 June 2006

www.cs.man.ac.uk/∼pt/ASD

http://www.cs.man.ac.uk/~pt/ASD/index.pdf

Russian Recursive AnalysisThe recursive real number a : R is one for whichthere is a program that, given k :N as input,yields p, q : Qwith p < a < q and q − p < 2−k.The recursive real line is the set of all such a : R.

Using some standard recursion theory...there exists a singular cover of R, i.e. a recursively enumerablesequence of intervals (pn, qn) ⊂ R with pn < qn : Q such thatI each recursive real number a lies in some interval (pn, qn),I but

∑n qn − pn < 1.

There is no finite subcover of I ≡ [0, 1].Measure theory also goes badly wrong.

One solution: Weihrauch’s Type Two EffectivityConsider all real numbers.Represent them (for example) by signed binary expansions

a =+∞∑

k=−∞

dk · 2−k with dk ∈ {+1, 0,−1}.

Think of {. . . , 0, 0, 0, . . . , d−2, d−1, , d0, d1, d2, . . .} as a Turing tapewith finitely many nonzero digits to the left,but possibly infinitely many to the right.

Do real analysis in the usual way.

Do computation with the sequences of digits.

Klaus Weihrauch, Computable Analysis, Springer, 2000.Vasco Brattka, Peter Hertling, Martin Ziegler, ...

Another solution: Bishop’s Constructive AnalysisLive without the Heine–Borel theorem.

Compact = closed and totally bounded.(X is totally bounded if, for any ε > 0, there’s a finite set Sε ⊂ Xsuch that for any x ∈ X there’s s ∈ Sε with d(x, s) < ε.)

Errett Bishop, Foundations of Constructive Analysis, 1967

He developed remarkably much of analysis in a “can do” way,without dwelling on counterexamples that arise from wrongclassical definitions.Consistent with both Russian Recursive Analysis and ClassicalAnalysis. Uses Intuitionistic Logic (Brouwer, Heyting).Douglas Bridges, Hajime Ishihara, Mark Mandelkern, RayMines, Fred Richman, Peter Schuster, ...

No explicit computation, butthe issues that Constructive Analysis raisesare often the same ones that Numerical Analysts experience.

Another solution: Bishop’s Constructive AnalysisLive without the Heine–Borel theorem.

Compact = closed and totally bounded.(X is totally bounded if, for any ε > 0, there’s a finite set Sε ⊂ Xsuch that for any x ∈ X there’s s ∈ Sε with d(x, s) < ε.)

Errett Bishop, Foundations of Constructive Analysis, 1967He developed remarkably much of analysis in a “can do” way,without dwelling on counterexamples that arise from wrongclassical definitions.Consistent with both Russian Recursive Analysis and ClassicalAnalysis. Uses Intuitionistic Logic (Brouwer, Heyting).Douglas Bridges, Hajime Ishihara, Mark Mandelkern, RayMines, Fred Richman, Peter Schuster, ...

No explicit computation, butthe issues that Constructive Analysis raisesare often the same ones that Numerical Analysts experience.

Disadvantages of these methodsPoint-set topology and recursion theory separately arecomplicated subjects that lack conceptual structure.

Together, they give pathological results.

Intuitionism makes things even worse —the natural relationship between open and closed subspacesis replaced by double negation.

Category theory can do better than this!

Some topology — the Sierpinski space

Classically, it’s just(�•

).

For every open subspace U ⊂ X

there’s a unique continuous function φ : X→(�•

)for which U = φ−1( � ).

U > �

X∨

.....................φ>(�•

)∨This is a bijective correspondence.

Some topology — the Sierpinski space

Classically, it’s just(�•

).

For every closed subspace C ⊂ X

there’s a unique continuous function φ : X→(�•

)for which C = φ−1(•).

C > •

X∨

.....................φ>(�•

)∨This is a bijective correspondence too.

The Sierpinski spaceFor every open subspace C ⊂ Xthere’s a unique continuous function φ : X→ Σfor which U = φ−1(>)

For every closed subspace C ⊂ Xthere’s a unique continuous function φ : X→ Σfor which C = φ−1(⊥).

There is a three-way correspondence.It’s not set-theoretic complementation.It doesn’t involve double negation or excluded middle.It’s topology, not set theory.

Relative containment of open subspacesLet σ, α, β be propositions (terms of type Σ) with parametersx1 : X1, ..., xk : Xk.They define open subspaces of Γ.The correspondence is supposed to be bijective.So they should satisfy a Gentzen-style rule of inference:

Γ, σ⇔ > ` α ⇒ β====================Γ ` σ ∧ α ⇒ β

in which the top line means

within the open subspace of Γ defined by σ,the open subspace defined by αis contained in the open subspace defined by β.

and the bottom line meansthe intersection of the open subspaces defined by σ and αis contained in that defined by β.

Relative containment of closed subspacesLet σ, α, β be propositions (terms of type Σ) with parametersx1 : X1, ..., xk : Xk.They define closed subspaces of Γ.The correspondence is supposed to be bijective.So they should satisfy a Gentzen-style rule of inference:

Γ, σ⇔ ⊥ ` α ⇒ β====================Γ ` α ⇒ σ ∨ β

in which the top line means

within the closed subspace of Γ defined by σ,the closed subspace defined by αcontains the closed subspace defined by β.

and the bottom line meansthe intersection of the closed subspaces defined by σ and βis contained in that defined by α.

The Euclidean & Phoa PrinciplesThe Gentzen-style rule for open subspaces,

Γ, σ⇔ > ` α ⇒ β====================Γ ` σ ∧ α ⇒ β

with α ≡ F> and β ≡ σ ∧ Fσ gives the Euclidean principle

σ ∧ F> ⇐⇒ σ ∧ Fσ.

Combining this with monotonicity, α⇒ β ` Fα⇒ Fβ,and the Gentzen-style rule for closed subspaces,we obtain the Phoa principle,

Fσ ⇐⇒ F⊥ ∨ σ ∧ F>.

Paul Taylor, Geometric and Higher Order Logic, 2000.

The topology as a function space

The topology on X is the set of functions X→ Σ ≡(�•

).

Function spaces X→ Y have a (compact–open) topology too.But it’s only well behaved when X is locally compact.Ralph Fox, Topologies on function spaces, 1945.To prove this, the critical case is Y ≡ Σ.Then X→ Σ carries the Scott topology.

Dana Scott, Continuous Lattices, 1972.

Compactness and Scott continuityA function F : L1 → L2 between complete latticesis Scott continuous iff it preserves directed joins.For example, let K ⊂ X be any subspaceand F : (X→ Σ)→ Σ the function for whichF(U) = > if K ⊂ U and ⊥ otherwise.Then F is Scott continuous iff K is compact.(This is just the “finite open subcover” definition in anotherform.)

Popularised by Martın Escardo, Synthetic Topology, 2004.

In set theory ∃ satisfies the Frobenius law,

∃x. σ ∧ φ(x) ⇐⇒ σ ∧ ∃x. φx

In topology ∀ also satisfies the dual Frobenius law

∀x. σ ∨ φ(x) ⇐⇒ σ ∨ ∀x. φx

Japie Vermeulen, Proper maps in locale theory, 1994.The Frobenius law for ∃ is a special case of the Euclideanprinciple, with

Fσ ≡ ∃x. σ ∧ φx.

Compactness and Scott continuityA function F : L1 → L2 between complete latticesis Scott continuous iff it preserves directed joins.For example, let K ⊂ X be any subspaceand ∀X : (X→ Σ)→ Σ the function for which∀X(U) = > if K ⊂ U and ⊥ otherwise.Then ∀X is Scott continuous iff K is compact.In set theory ∃ satisfies the Frobenius law,

∃x. σ ∧ φ(x) ⇐⇒ σ ∧ ∃x. φx


∀x. σ ∨ φ(x) ⇐⇒ σ ∨ ∀x. φx


Fσ ≡ ∃x. σ ∧ φx.

Compactness and Scott continuityA function F : L1 → L2 between complete latticesis Scott continuous iff it preserves directed joins.For example, let K ⊂ X be any subspaceand ∀X : (X→ Σ)→ Σ the function for which∀X(U) = > if K ⊂ U and ⊥ otherwise.Then ∀X is Scott continuous iff K is compact.In set theory and topology ∃ satisfies the Frobenius law,

∃x. σ ∧ φ(x) ⇐⇒ σ ∧ ∃x. φx


∀x. σ ∨ φ(x) ⇐⇒ σ ∨ ∀x. φx

Japie Vermeulen, Proper maps in locale theory, 1994.

The Frobenius law for ∃ is a special case of the Euclideanprinciple, with

Fσ ≡ ∃x. σ ∧ φx.

Compactness and Scott continuityA function F : L1 → L2 between complete latticesis Scott continuous iff it preserves directed joins.For example, let K ⊂ X be any subspaceand ∀X : (X→ Σ)→ Σ the function for which∀X(U) = > if K ⊂ U and ⊥ otherwise.Then ∀X is Scott continuous iff K is compact.In set theory ∃ satisfies the Frobenius law,

∃x. σ ∧ φ(x) ⇐⇒ σ ∧ ∃x. φx


∀x. σ ∨ φ(x) ⇐⇒ σ ∨ ∀x. φx


Fσ ≡ ∃x. σ ∧ φx.

This is still not enough to axiomatise topologyConsider the category Dcpo of posets with directed joins.It has all limits, colimits and function-spaces.

The Dedekind and Cauchy reals can be defined.

They carry the discrete order, and the discrete topology.

I ≡ [0, 1] and Cantor space are not compact.

This is just as bad as Russian Recursive Analysis.

Stone Duality and LocalesMarshall Stone, 1934: topology is dual to algebra.

The topology on X is an algebraic structure (finite meets andinfinitary joins).Continuous functions X→ Y correspond bijectively tohomomorphisms from topology on Y to topology on X.

Locale theory redefines topology as algebra.Peter Johnstone, Stone Spaces, CUP, 1983.

Eliminates many of the uses of the Axiom of Choice that plaguepoint-set topology.

Can be defined for sheaves,and satisfies the Heine–Borel theorem.

Abstract Stone DualityLocale theory still uses algebras with sets as carriers.

“What if” the algebras (topologies) are spaces too?

In category theory we may define algebras over any categorywe please, using a monad.

The category of topologies is Sop,the dual of the category S of “spaces”.

It’s also a category of algebras for amonad on S.

Sop

S

Σ(−)

∧

a Σ(−)

∨

Paul Taylor, 1993.Inspired by Robert Pare, Colimits in topoi, 1974.

Some generalised abstract nonsenseJon Beck (1966) characterised monadic adjunctions:

I Σ(−) : Sop→ S reflects invertibility,

i.e. if Σf : ΣY � ΣX then f : X � Y, andI Σ(−) : Sop

→ S creates Σ(−)-split coequalisers.

Category theory is a strong drug —it must be taken in small doses.

As in homeopathy (?),it gets more effective the more we dilute it!

Diluting Beck’s theorem (first part)If Σf : ΣY � ΣX then f : X � Y.

X is the equaliser of

X >ηX

> Σ2X ≡ ΣΣX

ηΣ2X >

Σ2ηX> Σ

4X

where ηX : x 7→ λφ. φx.

Diluting Beck’s theorem (first part)There’s an equivalent type theory for general spaces X.P : ΣΣ

Xis prime if

Γ, F : Σ3X ` F P = P(λx. F (λφ. φx)

).

This says that the composites

ΓP> ΣΣ

X >> Σ

4X

are equal.

So we should have a map Γ→ X.

Diluting Beck’s theorem (first part)P : ΣΣ

Xis prime if Γ, F : Σ3X ` F P = P

(λx. F (λφ. φx)

).

Γ ` P : ΣΣX

P is prime

Γ ` focus P : Xfocus I

Γ ` P : ΣΣX

P is prime

Γ, φ : ΣX` φ(focus P) = Pφ : Σ

focus β

Γ ` a, b : X Γ, φ : ΣX` φa = φb

Γ ` a = bT0

The definition thunk a = ηX(a) = λφ. φa serves as theelimination rule for focus. Using this, equivalent ways ofwriting the focus β and η (T0) rules are

thunk (focus P) = P and focus (thunk x) = x,

where P is prime.

Diluting Beck’s theorem (first part)For X ≡N this is definition by descriptionand general recursion.Paul Taylor, Sober Spaces and Continuations, 2002.

For X ≡ R it is Dedekind completeness.

Diluting Beck’s theorem (second part)Σ(−) : Sop

→ S creates Σ(−)-split coequalisers.

This means that (certain) subspaces exist,and they have the subspace topology.

E > > X >> Y

ΣIφ<....

............

.

φ >

Every open subspace of E is the restriction of one of X,in a canonical way.

There’s a corresponding type theory.Paul Taylor, Subspaces in Abstract Stone Duality, 2002.

It can be used to develop an abstract, finitary axiomatisation ofthe “way below” relation for continuous lattices.Paul Taylor, Computably based locally compact spaces, 2006.

Diluting Beck’s theorem (second part) even furtherΣ(−) : Sop

→ S creates Σ(−)-split coequalisers.In particular, the Dedekind reals can be expressed as anequaliser

R > > ΣQ × ΣQ >> Y

Σ <......

...........

>

where, classically, the map I takesan open subspace O ⊂ R to the open subspace

{(D,U) ∈ ΣQ × ΣQ | ∃d,u : Q. d ∈ D ∧ u ∈ U ∧ [d,u] ⊂ O}.

The idempotent on ΣΣQ×ΣQ can be defined just using rationals.

R can be defined abstractly from this.It satisfies the Heine–Borel theorem!



R > > ΣQ × ΣQ >> Y

Σ <......

...........

>




R can be defined abstractly from this.

It satisfies the Heine–Borel theorem!



R > > ΣQ × ΣQ >> Y

Σ <......

...........

>




R can be defined abstractly from this.It satisfies the Heine–Borel theorem!

Natural axioms for the realsThe arithmetic operations: 0, 1, +, −, ×, ÷.The arithmetic relations: <, >, ,.(Not ≤, ≥ and = since they’re nottopologically open or computationally observable)Geometric logic: >, ⊥, ∧, ∨, ∃N, ∃R.(Not ¬ or⇒ since they would solve the halting problem.)Primitive recursion overN.Dedekind completeness and the Heine–Borel theorem.Universal quantification (∀) is defined over compact subspaces.

Andrej Bauer and Paul Taylor, The Dedekind Reals in ASD, 2005.

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